1. **State the problem:** We want to approximate the value of $y$ at $x=1.1$ given that $y=1.2$ at $x=1$ and the differential equation $$\frac{dy}{dx} = 3x + y^2.$$ We will use Runge's method (specifically the second-order Runge-Kutta method) to find this approximation. 2. **Recall the formula for the second-order Runge-Kutta method:** Given step size $h$, current point $(x_n, y_n)$, the next value $y_{n+1}$ is approximated by $$ y_{n+1} = y_n + h \cdot k_2, $$ where $$ k_1 = f(x_n, y_n), \quad k_2 = f\left(x_n + \frac{h}{2}, y_n + \frac{h}{2} k_1\right), $$ and $f(x,y) = \frac{dy}{dx}$. 3. **Identify given values:** - Initial point: $x_0 = 1$, $y_0 = 1.2$ - Step size: $h = 1.1 - 1 = 0.1$ - Differential function: $f(x,y) = 3x + y^2$ 4. **Calculate $k_1$:** $$ k_1 = f(1, 1.2) = 3(1) + (1.2)^2 = 3 + 1.44 = 4.44. $$ 5. **Calculate $k_2$:** $$ x_1 = 1 + \frac{0.1}{2} = 1.05, $$ $$ y_1 = 1.2 + \frac{0.1}{2} \times 4.44 = 1.2 + 0.222 = 1.422, $$ $$ k_2 = f(1.05, 1.422) = 3(1.05) + (1.422)^2 = 3.15 + 2.022 = 5.172. $$ 6. **Calculate $y_{1.1}$:** $$ y_{1.1} = 1.2 + 0.1 \times 5.172 = 1.2 + 0.5172 = 1.7172. $$ **Final answer:** The approximate value of $y$ at $x=1.1$ using Runge's method is **$1.7172$**.